arXiv:q-alg/9610031v1 25 Oct 1996 CPTH-S458.0696 q-alg/9610031 June 96 On U h (sl (2)), U h (e(3)) and their Representations B. Abdesselam 1 , A. Chakrabarti 2 and R. Chakrabarti 3 4 Centre de Physique Th´ eorique, Ecole Polytechnique, 91128 Palaiseau Cedex, France. Abstract By solving a set of recursion relations for the matrix elements of the U h (sl(2)) generators, the finite dimensional highest weight representations of the algebra were obtained as factor representations. Taking a nonlinear combination of the generators of the two copies of the U h (sl(2)) algebra, we obtained U h (so(4)) algebra. The latter, on contraction, yields U h (e(3)) algebra. A nonlinear map of U h (e(3)) algebra on its classical analogue e(3) was obtained. The inverse mapping was found to be singular. It signifies a physically interesting situation, where in the momentum basis, a restricted domain of the eigenvalues of the classical operators is mapped on the whole real domain of the eigenvalues of the deformed operators. 1 abdess@orphee.polytechnique.fr 2 chakra@orphee.polytechnique.fr 3 Permanent address: Department of Theoretical Physics, University of Madras, Guindy Campus, Madras- 600025, India 4 Laboratoire Propre du CNRS UPR A.0014